Finite Group Databasebeta

← #121824

Subgroups — #121824

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13966C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C2×#1294C212yesmaximal
4C2×#1289C212yesmaximal
5#7907C211yesmaximal, characteristic
6C22×C22⋊C8C2212yes—
7C2×#380C2212yes—
8C2×#380C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C2211yescharacteristic
11C2×#342—21no—
12C2×#342—21no—
13#1442—22no—
14#1434C412yescomplemented
15#1434C412yescomplemented
16C22⋊#95—21no—
17C22⋊#95—21no—
18#1294—24no—
19#1289—24no—
20#1280—22no—
21C24×C4—21no—
22C24×C4C2311yescharacteristic
23C23×C8—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42C2×C412yes—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8Q814yes—
37#380—22no—
38#380—22no—
39#380—22no—
40#380—22no—
41#380—22no—
42#380D414yes—
43C4×C22⋊C4—22no—
44C4×C22⋊C4—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—21no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95C2×C412yes—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D412yes—
57#342—42no—
58#342—42no—
59C25C2×D411yescharacteristic
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—44no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—82no—
70C23×C4—82no—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D412yes—
76C23×C4C2×Q812yes—
77C23×C4#4112yes—
78C23×C4#4112yes—
79C23×C4#4112yes—
80C23×C4#4112yes—
81C22×C8—24no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—84no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4—84no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—82no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#95—24no—
117#95—24no—
118#95—22no—
119#95—22no—
120#95—24no—
121#95—44no—
122#95—44no—
123#95—41no—
124#95—42no—
125#95—41no—
126#95—82no—
127#95—82no—
128C24—42no—
129C24—42no—
130C24C2×C22⋊C411yescharacteristic
131C24#12112yes—
132C24#12312yes—
133C24#12411yescharacteristic
134C24C4⋊Q811yescharacteristic
135C24#13512yes—
136C24#13512yes—
137C24AGL(1,ℤ/8)11yescharacteristic
138C24AGL(1,ℤ/8)11yescharacteristic
139C24#13711yescharacteristic
140C24#13711yescharacteristic
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—48no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—44no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—88no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4—84no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—84no—
185C22×C4—84no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4#11711yescharacteristic
189C22×C4#11711yescharacteristic
190C22×C4C4×D412yes—
191C22×C4C4×Q812yes—
192C22×C4C22⋊Q812yes—
193C22×C4C22⋊Q812yes—
194C22×C4C22⋊Q812yes—
195C22×C4#12312yes—
196C22×C4#12312yes—
197C22×C4#12312yes—
198C22×C4#12411yescharacteristic
199C22×C4#12512yes—
200C22×C4#12512yes—
201C22×C4#12612yes—
202C22×C4#12612yes—
203C22×C4#12711yescharacteristic
204C2×C8—44no—
205C2×C8—84no—
206C2×C8—84no—
207C2×C8—84no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—82no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—82no—
223C42—82no—
224C23—22no—
225C23—24no—
226C23—22no—
227C23—24no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—41no—
232C23—44no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—41no—
237C23—44no—
238C23—42no—
239C23—42no—
240C23#38111yescharacteristic
241C23#39011yescharacteristic
242C23#39111yescharacteristic
243C23#39712yes—
244C23#39812yes—
245C23#41611yescharacteristic
246C23#41611yescharacteristic
247C23#41711yescharacteristic
248C23#41711yescharacteristic
249C23#46312yes—
250C23#46312yes—
251C23#46412yes—
252C23#46412yes—
253C23#48212yes—
254C23#48212yes—
255C23#48512yes—
256C23#48512yes—
257C23#48812yes—
258C23#49011yescharacteristic
259C23#49111yescharacteristic
260C23#49912yes—
261C23#50111yescharacteristic
262C23#50111yescharacteristic
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—88no—
292C2×C4—88no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—164no—
312C2×C4—164no—
313C2×C4#38712yes—
314C2×C4#38712yes—
315C2×C4#39912yes—
316C2×C4#40012yes—
317C8—84no—
318C22—22no—
319C22—22no—
320C22—24no—
321C22—24no—
322C22—22no—
323C22—22no—
324C22—24no—
325C22—24no—
326C22—21no—
327C22—22no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22#144511yescharacteristic
345C22#182511yescharacteristic
346C22#182511yescharacteristic
347C22#182712yes—
348C22#182812yes—
349C22#198311yescharacteristic
350C22#198311yescharacteristic
351C22#199111yescharacteristic
352C22#199211yescharacteristic
353C22#207312yes—
354C22#207412yes—
355C22#208512yes—
356C22#208512yes—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—84no—
362C4—82no—
363C4—82no—
364C4—161no—
365C4—161no—
366C4—164no—
367C2—21no—
368C2—22no—
369C2—21no—
370C2—22no—
371C2—22no—
372C2—22no—
373C2—21no—
374C2—21no—
375C2#876212yes—
376C2#876312yes—
377C2#1134711yescharacteristic
378C2#1135111yescharacteristic
379C2#1135811yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. #1434C4semidirect
15. #1434C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#139662
#108462
C2×#12942
C2×#12892
#79072